Diffraction of the Ulam sequence: covariances and outliers

We look at the Ulam sequence 1, 2, 3, 4, 6, 8, 11, … through its diffraction measure, the Fourier transform of its autocorrelation. If the autocorrelation exists and the terms have an angular distribution g with respect to Steinerberger's frequency α, the diffraction is the sum of Bragg peaks at the multiples of α/2π, with intensities |ĝ(m)|², and a remainder whose Fourier coefficients are the covariances c(k); if Σ c(k)² < ∞, the remainder is absolutely continuous with an L² density. This step is standard and close to a reformulation of the assumptions. Our interest is in the covariances. Following Gibbs, call outliers the terms whose angle lies outside the middle third of the circle. We prove that if the outliers have density zero, the covariance at an outlier lag e is at most the angular width ℓ(e) through which e can contribute, so that the outlier lags contribute a finite amount to Σ c(k)² whenever Σ_e ℓ(e)² < ∞, a quantitative form of Gibbs's conjecture. We then study the covariances numerically on the integers below 2^27, for a fixed value of α. The background is far from the flat background of an independent thinning with the same peaks, and its shape is dominated by the single covariance c(2). Gibbs's independence formula, fed only with α and the outliers up to 10^5, reproduces the normalised peak heights to within 0.004, and the same formula, with the outliers up to 10^4, predicts the covariances at the outlier lags (correlation 0.997). Over shorter ranges the angular distribution fluctuates mainly by rotations, a phason-like mode in the language of quasicrystals: the phases of the harmonics move roughly in the ratio 1:2:…:6, the rotation resembles the slowest mode of the linearised Gibbs equation (eigenvalue 0.841), it is cut off only at the edges of the middle third, and new outliers tend to appear at its leading edge. These are numerical observations; the mechanism we propose for them is a hypothesis. The numerical statements concern a finite range. The proofs are short and written out in full, but their validity is left open to external verification; see the note at the end of the paper. The accompanying archive contains the programs and the recorded outputs.